Foundation Logic As Functional Equation Logic Scale Invariant L To Real

Scale invariance, transported

Scale invariance is a symmetry property of a comparison operation. It says that multiplying both inputs by the same positive factor leaves the output unchanged. For example, if a comparison of lengths gives the same result whether you measure in meters or kilometers, that comparison is scale invariant. The declaration scaleInvariantL_to_real establishes that this property, when it holds for comparisons defined on a special number system, also holds for the corresponding comparisons on ordinary real numbers.

The special number system is a constructed set of numbers that the framework uses as an intermediate layer. The declaration shows that the property is preserved when moving between these two systems. This is a transport theorem: it takes a statement true in one setting and shows it remains true in another, more familiar setting. The proof works by translating the inputs and outputs through a conversion function and checking that the defining equation of scale invariance still holds.

In Recognition Science, this transport is part of a larger chain. The framework models comparison operations as a kind of cost or ledger, a discrete record of events. The scale invariance property is one of several structural conditions that, together, force a specific functional form for this cost. The declaration here ensures that this structural condition, when stated in the intermediate number system, is not lost when moving to the real numbers where the main theorems are proved.

The declaration does not claim that scale invariance holds for any particular comparison operation. It is a conditional statement: if a comparison satisfies scale invariance in the special system, then its transported version satisfies it in the real numbers. It also does not claim that scale invariance alone forces any particular form of the cost function. That conclusion requires the full set of conditions, including identity, symmetry, and non-triviality, as proved elsewhere in the framework's library of formal theorems.

theorem scaleInvariantL_to_real (C : ComparisonOperatorL) (h : ScaleInvariantL C) :
    LogicAsFunctionalEquation.ScaleInvariant (transportComparison C) := by
  intro x y lam hx hy hlam
  unfold transportComparison
  have hxL : (0 : LogicReal) < fromReal x := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hx
  have hyL : (0 : LogicReal) < fromReal y := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hy
  have hlamL : (0 : LogicReal) < fromReal lam := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hlam
  have hmulx : fromReal lam * fromReal x = fromReal (lam * x) := by
    rw [eq_iff_toReal_eq]; simp [toReal_fromReal]
  have hmuly : fromReal lam * fromReal y = fromReal (lam * y) := by
    rw [eq_iff_toReal_eq]; simp [toReal_fromReal]
  have hL := h (fromReal x) (fromReal y) (fromReal lam) hxL hyL hlamL
  rw [hmulx, hmuly] at hL
  exact congrArg toReal hL

theorem scaleInvariantL_to_real (C : ComparisonOperatorL) (h : ScaleInvariantL C) :
    LogicAsFunctionalEquation.ScaleInvariant (transportComparison C) := by
  intro x y lam hx hy hlam
  unfold transportComparison
  have hxL : (0 : LogicReal) < fromReal x := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hx
  have hyL : (0 : LogicReal) < fromReal y := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hy
  have hlamL : (0 : LogicReal) < fromReal lam := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hlam
  have hmulx : fromReal lam * fromReal x = fromReal (lam * x) := by
    rw [eq_iff_toReal_eq]; simp [toReal_fromReal]
  have hmuly : fromReal lam * fromReal y = fromReal (lam * y) := by
    rw [eq_iff_toReal_eq]; simp [toReal_fromReal]
  have hL := h (fromReal x) (fromReal y) (fromReal lam) hxL hyL hlamL
  rw [hmulx, hmuly] at hL
  exact congrArg toReal hL

What this page does not claim

Scale invariance holds for any particular comparison operation. Scale invariance alone forces any specific form of the cost function. The declaration proves anything about the uniqueness of the cost function without the other structural conditions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/LogicAsFunctionalEquationLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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